128 5. COUNTINGrevolution in 26,000 years) in the direction opposite to the Sun's motion along the
ecliptic, a tropical year is about 20 minutes shorter than a sidereal year. Would you
expect the flooding of the Nile to be synchronous with the tropical year or with the
sidereal year? If the flooding is correlated with the tropical year, how long would
it take for the heliacal rising of Sirius to be one day out of synchronicity with the
Nile flood? If the two were synchronous 4000 years ago, how far apart would they
be now, and would the flood occur later or earlier than the heliacal rising of Sirius?5.5. How many Tzolkin cycles are there in a Calendar Round?5.6. The pattern of leap-year days in the Gregorian calendar has a 400-year cycle.
Do the days of the week also recycle after 400 years?
5.7. (The. revised Julian calendar) The Gregorian calendar bears the name of the
Pope who decreed that it should be used. It was therefore adopted early in many
countries with a Catholic government, somewhat later in Anglican and Protestant
countries. Countries that are largely Orthodox in faith resisted this reform until
the year 1923, when a council suggested that century years should be leap years
only when they leave a remainder of 2 or 6 when divided by 9. (This reform was not
mandated, but was offered as a suggestion, pending universal agreement among all
Christians on a date for Easter.) This modification would retain only two-ninths of
the century years as leap years, instead of one-fourth, as in the Gregorian calendar.
What is the average number of days in a year of this calendar? How does it compare
with the actual length of a year? Is it more or less accurate than the Gregorian
calendar?5.8. In constructing a calendar, we encounter the problem of measuring time.
Measuring space is a comparatively straightforward task, based on the notion of
congruent lengths. One can use a stick or a knotted rope stretched taut as a
standard length and compare lengths or areas using it. Two lengths are congruent
if each bears the same ratio to the standard length. In many cases one can move the
objects around and bring them into coincidence. But what is meant by congruent
time intervals? In what sense is the interval of time from 10:15 to 10:23 congruent
to the time interval from 2:41 to 2:49?
5.9. It seems clear that the decimal place-value system of writing integers is po-
tentially infinite; that is there is no limit on the size of number that can be written
in this system. But in practical terms, there is always a largest number for which
a name exists. In ordinary language, we can talk about trillions, quadrillions,
quintillions, sextillions, septillions, octillions, and so on. But somewhere before
the number 10^60 is reached, most people (except Latin scholars) will run out of
names. Some decades ago, a nephew of the American mathematician Edward Kas-
ner (1878-1955) coined the name googol for the number 10100 , and later the name
googolplex for 10^10. This seems to be the largest number for which a name exists
in English. Does there exist a positive integer for which no name could possibly
be found, not merely an integer larger than all the integers that have been or will
have been named before the human race becomes extinct? Give a logical argument
in support of your answer. (And, while you arc at it, consider what is meant by
saying that an integer "exists.")